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Restriction of a marked ideal to a smooth subvariety and its blow-ups

Statement

Assume AC (The Axiom of Choice) for the regular-parameter and blowup suppliers.

Let $(I, E, μ)$ be a marked ideal of maximal order on the smooth $K$ -scheme $X$ , and let $S \subseteq X$ be a regular closed subscheme having SNC with $E$ and not contained in $supp (I, E, μ)$ (Marked ideals and their support, Simple normal crossings divisors and simultaneous normal crossings position). For restriction to $S$ , work componentwise and omit from the restricted boundary the members containing that component of $S$ ; retain the restrictions of the other members in their original order. Here SNC position for $S$ means that its ideal is generated by a subset of parameters compatible with the boundary equations. This convention is required, for example, when $S$ is itself a boundary stratum: a divisor containing $S$ does not restrict to a Cartier divisor on it.

Then $supp (I, μ) \cap S \subseteq supp (I ∣_{S}, μ)$ , where $I ∣_{S}$ is the pullback ideal on $S$ (Coherent module sheaves). If $C \subseteq supp (I, μ) \cap S$ is a regular center with SNC with $E$ , $σ ⁣ : X^{'} \to X$ is the blowup and $S^{'} \subseteq X^{'}$ is the strict transform of $S$ (Strict transform of a closed subscheme), then $σ^{c} ((I, μ) ∣_{S}) = (σ^{c} (I, μ)) ∣_{S^{'}} .$ Moreover, for any multiple test blow-up $(X_{i})$ of $(I, μ)$ all of whose centers lie in the strict transforms $S_{i}$ of $S$ , the restrictions $σ_{i} ∣_{S_{i}}$ define a multiple test blow-up $(S_{i})$ of $(I, μ) ∣_{S}$ and $[(I, μ) ∣_{S}]_{i} = (I_{i}, μ) ∣_{S_{i}}$ for every $i$ .

Facts & Assumptions

Given: A marked ideal $(I, E, μ)$ on a smooth $K$ -scheme $X$ whose support does not contain a smooth subvariety $S \subseteq X$ that has SNC with $E$ ; a blowup $σ ⁣ : X^{'} \to X$ with center $C \subseteq supp (I, E, μ) \cap S$ ; the strict transform $S^{'} \subseteq X^{'}$ of $S$ .

[F1]

Marked ideals and their support: the restriction of the marked ideal is $(I, μ) ∣_{S} = (I \cdot O_{S}, μ)$ , with support ${x \in S : {ord}_{x} (I O_{S}) \geq μ}$ .

[F2]

Order of an ideal sheaf at a point: order is defined by containment of stalks in powers of the maximal ideal; for $x \in S$ the maximal ideal of $O_{S, x}$ is the image of $m_{X, x}$ , so $I_{x} \subseteq m_{X, x}^{μ}$ implies $I_{x} O_{S, x} \subseteq m_{S, x}^{μ}$ ; restriction can only raise the order.

[F3]

Multiple test blow-ups, controlled transforms and resolutions of marked ideals, Exceptional subscheme of a blowup: the controlled transform of a section $f \in I (U)$ is $f^{'} = y^{- μ} σ^{*} (f)$ for a local equation $y$ of the exceptional divisor $D$ ; the controlled transform of the marked ideal is generated by the $f^{'}$ .

[F4]

Strict transform of a closed subscheme, Blowup of a scheme along an ideal sheaf: in coordinates $x_{1}, \dots, x_{k}$ defining $S$ and $y_{1}, \dots, y_{n - k}$ along it at a point of $C$ , with the center described by $x_{1}, \dots, x_{k}, y_{1}, \dots, y_{m}$ , the chart of the blowup has coordinates $x_{i}^{'} = x_{i} / y_{m}$ $(i \leq k)$ , $y_{j}^{'} = y_{j} / y_{m}$ $(j < m)$ , $y_{m}^{'} = y_{m}$ , $y_{j}^{'} = y_{j}$ $(j > m)$ , and the strict transform $S^{'}$ is described by $x_{1}^{'} = \dots = x_{k}^{'} = 0$ with $y_{m}^{'}$ a local equation of the exceptional divisor of $S^{'} \to S$ .

[F5]

embedding dimension and regular local ring, Simple normal crossings divisors and simultaneous normal crossings position: $S$ is smooth with SNC with $E$ , so the coordinates can be chosen adapted both to $S$ and to $E$ ; after omitting the members containing a component of $S$ , the remaining restrictions are again a family in simultaneous SNC position. The same omission convention applies at every stage.

Proof

1.1F1F2

The first inclusion. Let $x \in supp (I, μ) \cap S$ , so $I_{x} \subseteq m_{X, x}^{μ}$ . Applying the ring map $O_{X, x} \to O_{S, x}$ gives $I_{x} O_{S, x} \subseteq m_{X, x}^{μ} O_{S, x} = m_{S, x}^{μ}$ , so ${ord}_{x} (I \cdot O_{S}) \geq μ$ and $x \in supp ((I, μ) ∣_{S})$ . Hence $supp (I, μ) \cap S \subseteq supp ((I, μ) ∣_{S})$ .

1.2F3F4F5algebra

Work at a point of $S^{'}$ over $C$ . By the parameter-generation theorem, the center ideal is $(x_{1}, \dots, x_{k}, y_{1}, \dots, y_{m})$ , where $S$ is defined by the $x$ 's. On a chart indexed by an $x_{i}$ , saturation makes the strict transform of $S$ empty. On a chart indexed by $a = y_{j}$ , it is defined by $x_{i} / a = 0$ , and its chart is precisely the corresponding chart of ${Bl}_{C} S$ . The exceptional equation on $S^{'}$ is $a ∣_{S^{'}}$ , a nonzerodivisor. Thus for every generator $f$ of $I$ , restriction of $a^{- μ} σ^{*} f$ to $S^{'}$ equals $(a ∣_{S^{'}})^{- μ} (σ ∣_{S^{'}})^{*} (f ∣_{S})$ . This proves the transform identity on every nonempty chart without assuming a power-series expansion; the empty charts have no stalk to check. These identities glue, and the remaining restricted boundary has SNC by [F5].

2.1F1F5step 1.2∎

Iteration along a multiple test blow-up. Suppose $(X_{i})_{0 \leq i \leq r}$ is a multiple test blow-up of $(I, μ)$ with every center $C_{i}$ contained in the strict transform $S_{i}$ of $S$ . By induction on $i$ , step 1.2 applied to the restricted marked ideal $(I_{i}, μ) ∣_{S_{i}}$ and the blowup $σ_{i + 1}$ with center $C_{i} \subseteq S_{i}$ gives $[(I, μ) ∣_{S}]_{i + 1} = σ_{i + 1}^{c} ((I_{i}, μ) ∣_{S_{i}}) = ((I_{i + 1}, μ) ∣_{S_{i + 1}})$ ; moreover $C_{i}$ , being also a center for the restricted marked ideal with SNC with $E_{i} ∣_{S_{i}}$ by [F5], makes $(S_{i})$ a multiple test blow-up of $(I, μ) ∣_{S}$ with the same transform rule. The base case $i = 0$ is the identity. This proves the final assertion for every length, and in particular the equality $[(I, μ) ∣_{S}]_{i} = (I_{i}, μ) ∣_{S_{i}}$ stated in the lemma.

Remarks

The hypothesis that $S$ is not contained in $supp (I, μ)$ is used only to keep the two suppressed-locus readings apart; the computation of step 1.2 uses no such hypothesis, and the first inclusion of step 1.1 is unconditional. The empty case $S = \emptyset$ makes all statements vacuous.
The identity is the source's Lemma 2.10.3 and is the restriction calculus on which the coefficient-ideal lemmas below are built.

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Sources

Jaroslaw Wlodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005) 779-822; author's arXiv version math/0401401 (28 pp., dated October 25, 2018) (standard reference, not scraped)